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人気のある微分積分 >

limit as x approaches 0+of xln(tan(3x))

解

x \to 0 + lim (x ln (tan (3 x)))

解

0

解答ステップ

x \to 0 + lim (x ln (tan (3 x)))

ロピタル向けに書き換える

= x \to 0 + lim (\frac{ln ( tan ( 3 x ) )}{\frac{1}{x}})

ロピタルの定理を適用する

= x \to 0 + lim (\frac{3 sec ^{2} ( 3 x ) cot ( 3 x )}{- \frac{1}{x ^{2}}})

簡素化

= x \to 0 + lim (- 3 x^{2} sec^{2} (3 x) cot (3 x))

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= - 3 \cdot x \to 0 + lim (x^{2} sec^{2} (3 x) cot (3 x))

サイン, コサインで表わす

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{cos ( 3 x ) sin ( 3 x )})

\frac{x ^{2}}{cos ( 3 x ) sin ( 3 x )} = \frac{x ^{2}}{sin ( 3 x )} \cdot \frac{1}{cos ( 3 x )}

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 3 x )} \cdot \frac{1}{cos ( 3 x )})

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

不定形式の場合を除く

= - 3 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 3 x )}) \cdot x \to 0 + lim (\frac{1}{cos ( 3 x )})

x \to 0 + lim (\frac{x ^{2}}{sin ( 3 x )}) = 0

x \to 0 + lim (\frac{1}{cos ( 3 x )}) = 1

= - 3 \cdot 0 \cdot 1

簡素化

= 0

グラフ

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